In the field of robotic manipulation, the dexterous robotic hand serves as the quintessential end-effector, enabling robots to interact with and manipulate objects in complex, human-like ways. The primary and most critical function of a dexterous robotic hand is to achieve stable and reliable grasping. As robots find applications in increasingly diverse and unstructured environments—from industrial assembly to domestic assistance and healthcare—the planning of coordinated multi-fingered grasps becomes paramount. A fundamental challenge in this domain is determining the appropriate contact forces the fingers must apply. While intuition might suggest that a stronger grasp equates to a more secure one, this is not universally optimal. Excessive force can lead to the deformation or damage of fragile objects (e.g., ceramic vases, light bulbs, ripe fruit). Conversely, when grasping smooth objects, an overly strong grip can paradoxically increase the risk of slippage. On the other hand, applying insufficient force compromises stability, making the grasp vulnerable to failure under small external disturbances or due to inherent uncertainties in contact modeling and positioning.
Therefore, a superior grasp planning strategy for a dexterous robotic hand aims to apply the minimum necessary force to achieve stability. This minimizes energy consumption, reduces mechanical wear on the hand, and, most importantly, safeguards the manipulated object. This article presents a novel grasp planning algorithm centered on minimizing the normal components of the contact forces, which are primarily responsible for potential crushing or damaging effects. The planning is formulated as a constrained optimization problem, solved using an adaptive Differential Evolution algorithm, and evaluated using established grasp quality metrics.
The core of multi-fingered grasp analysis for a dexterous robotic hand lies in modeling the contacts and the force equilibrium. We consider the common and practical model of hard-finger contact with friction. At each contact point \(i\), the force applied by the finger, denoted as \(\mathbf{f}_i = (f_{ix}, f_{iy}, f_{iz})^T\), must lie inside the friction cone to prevent slip. This imposes a nonlinear constraint: the tangential force components must not exceed the product of the friction coefficient \(\mu_i\) and the normal force component. Furthermore, the normal force must be positive (pushing into the object, not pulling). Mathematically, for \(n\) contacts, this is expressed as:
$$
\Psi_i = f_{ix}^2 + f_{iy}^2 – \mu_i^2 f_{iz}^2 \leq 0, \quad f_{iz} \geq 0, \quad i = 1, 2, …, n
$$
For a grasp to be stable against arbitrary external wrenches (forces and torques), it must satisfy the condition of force closure. This means the set of contact forces achievable by the dexterous robotic hand can balance any external wrench applied to the object. The fundamental equation governing the static equilibrium of the grasped object is given by the grasp map:
$$
\mathbf{G} \mathbf{f}_{ext} = -\mathbf{F}_{ext}
$$
Where \(\mathbf{F}_{ext} = [\mathbf{f}_o, \mathbf{\tau}_o]^T \in \mathbb{R}^6\) is the external wrench (force \(\mathbf{f}_o\) and torque \(\mathbf{\tau}_o\)) acting on the object’s center of mass. \(\mathbf{f}_{ext} = [\mathbf{f}_1^T, \mathbf{f}_2^T, …, \mathbf{f}_n^T]^T \in \mathbb{R}^{3n}\) is the concatenated vector of all contact forces. \(\mathbf{G} \in \mathbb{R}^{6 \times 3n}\) is the grasp matrix, which projects contact forces from the local contact coordinates into the object coordinate frame. For a contact point \(i\) with position \(\mathbf{p}_i = (X_i, Y_i, Z_i)^T\) relative to the object frame and contact frame orientation angles \(\theta_{ix}, \theta_{iy}\), the corresponding block \(\mathbf{G}_i\) of the grasp matrix is derived from the adjoint transformation:
$$
\mathbf{G}_i = \begin{bmatrix}
\mathbf{R}_i & \mathbf{0} \\
\mathbf{\hat{p}}_i \mathbf{R}_i & \mathbf{R}_i
\end{bmatrix} \begin{bmatrix}
\mathbf{I}_{3\times3}\\
\mathbf{0}
\end{bmatrix} = \begin{bmatrix}
\mathbf{R}_i \\
\mathbf{\hat{p}}_i \mathbf{R}_i
\end{bmatrix}
$$
Here, \(\mathbf{R}_i\) is the rotation matrix from the contact frame to the object frame, and \(\mathbf{\hat{p}}_i\) is the skew-symmetric matrix representing the cross product for \(\mathbf{p}_i\). For a three-fingered dexterous robotic hand grasping a spherical object of radius \(R\), the explicit form for a single contact is:
$$
\mathbf{G}_i = \begin{bmatrix}
C\theta_{ix} & -S\theta_{ix}C\theta_{iy} & -S\theta_{ix}S\theta_{iy} \\
S\theta_{ix} & C\theta_{ix}C\theta_{iy} & -C\theta_{ix}S\theta_{iy} \\
0 & S\theta_{iy} & C\theta_{iy} \\
Y_i S\theta_{iy} – Z_i C\theta_{ix}C\theta_{iy} & Y_i C\theta_{iy} + Z_i C\theta_{ix}S\theta_{iy} & -X_i S\theta_{iy} – Z_i S\theta_{ix}C\theta_{iy} \\
-X_i S\theta_{iy} – Z_i S\theta_{ix}C\theta_{iy} & -X_i C\theta_{iy} – Z_i S\theta_{ix}S\theta_{iy} & X_i C\theta_{ix}C\theta_{iy} + Y_i S\theta_{ix}C\theta_{iy} \\
X_i C\theta_{ix}C\theta_{iy} + Y_i C\theta_{iy}S\theta_{ix} & Y_i S\theta_{ix}S\theta_{iy} – X_i C\theta_{ix}S\theta_{iy} & -Y_i C\theta_{ix} – Z_i S\theta_{ix}
\end{bmatrix}
$$
Where \(C\theta = \cos(\theta)\) and \(S\theta = \sin(\theta)\). The full grasp matrix is \(\mathbf{G} = [\mathbf{G}_1, \mathbf{G}_2, \mathbf{G}_3]\).
The contact forces are ultimately generated by the joint actuators of the dexterous robotic hand. Therefore, they are limited by the maximum joint torques \(\boldsymbol{\tau}_{max}\). This imposes another linear constraint derived from the hand’s Jacobian \(\mathbf{J}_i\) for finger \(i\):
$$
\mathbf{J}_i^T \mathbf{f}_i \leq \boldsymbol{\tau}_{max}, \quad i=1,2,…,n
$$
Traditional grasp planning often focuses on minimizing the total force magnitude or ensuring force closure with margin. However, inspired by human grasping behavior, we argue that the normal component of the contact force (\(f_{iz}\)) is the principal cause of potential damage due to “crushing.” The tangential components contribute primarily to generating stabilizing moments. Therefore, our proposed objective for the dexterous robotic hand is to minimize the sum of the normal contact forces (or a weighted sum for objects with non-uniform surface stiffness), while strictly satisfying all physical and stability constraints. This leads to the following nonlinear optimization problem:
$$
\begin{aligned}
& \underset{\mathbf{f}_{ext}, \mathbf{p}_i, \boldsymbol{\theta}_i}{\text{minimize}}
& & \sum_{i=1}^{n} \frac{1}{2} w_i f_{iz}^2 = \frac{1}{2} \mathbf{f}_z^T \mathbf{Q} \mathbf{f}_z \\
& \text{subject to}
& & \mathbf{G}(\mathbf{p}_i, \boldsymbol{\theta}_i) \mathbf{f}_{ext} = -\mathbf{F}_{ext} \quad \text{(Force Equilibrium)} \\
& & & f_{iz} \geq 0, \quad i = 1,…,n \quad \text{(Unilateral Contact)} \\
& & & \mu_i^2 f_{iz}^2 – f_{ix}^2 – f_{iy}^2 \geq 0, \quad i = 1,…,n \quad \text{(Friction Cone)} \\
& & & \boldsymbol{\tau}_{max} – \mathbf{J}_i^T \mathbf{f}_i \geq 0, \quad i = 1,…,n \quad \text{(Actuation Limits)} \\
& & & \boldsymbol{\theta}_{min} \leq \boldsymbol{\theta}_i \leq \boldsymbol{\theta}_{max} \quad \text{(Contact Orientation Bounds)} \\
& & & X_i^2 + Y_i^2 + Z_i^2 = R^2 \quad \text{(Contact Point on Sphere Surface)}
\end{aligned}
$$
Here, \(\mathbf{f}_z = [f_{1z}, f_{2z}, …, f_{nz}]^T\), \(\mathbf{Q}\) is a diagonal weighting matrix, and \(\boldsymbol{\theta}_i = [\theta_{ix}, \theta_{iy}]^T\). The decision variables are both the contact forces \(\mathbf{f}_{ext}\) and the contact locations/orientations \((\mathbf{p}_i, \boldsymbol{\theta}_i)\). For non-spherical objects, the surface constraint \(X_i^2 + Y_i^2 + Z_i^2 = R^2\) would be replaced by the appropriate implicit or parametric equation of the object’s surface.
Solving this constrained, nonlinear, and potentially non-convex problem requires a robust global optimization algorithm. We employ an Adaptive Differential Evolution (DE) algorithm, known for its simplicity, strong robustness, and fast convergence. DE operates on a population of candidate solutions, iteratively improving them through cycles of mutation, crossover, and selection. The adaptive mechanism tunes the algorithm’s key parameters (scaling factor \(F\) and crossover rate \(CR\)) based on historical success, enhancing performance. The process is summarized in the following table:
| Step | Operation | Mathematical Description |
|---|---|---|
| 1 | Initialization | Generate initial population of size \(NP\) with dimension \(D\). Set generation \(G=1\). |
| 2 | Termination Check | If stopping criteria met, output best solution. Else, continue. |
| 3 | Mutation | Create mutant vector: \(\mathbf{V}_{r,G} = \mathbf{X}_{r1,G} + F_r (\mathbf{X}_{r2,G} – \mathbf{X}_{r3,G})\). \(F_r\) is adaptively sampled from a Cauchy distribution. |
| 4 | Crossover | Create trial vector \(\mathbf{U}_{i,G}\) by mixing components from mutant \(\mathbf{V}_{i,G}\) and target \(\mathbf{X}_{i,G}\) based on adaptive rate \(CR_i\) (sampled from a Gaussian distribution). |
| 5 | Selection | Select the better vector between trial \(\mathbf{U}_{i,G}\) and target \(\mathbf{X}_{i,G}\) for the next generation: \(\mathbf{X}_{i,G+1} = \arg\min(f(\mathbf{U}_{i,G}), f(\mathbf{X}_{i,G}))\). Set \(G = G+1\), go to Step 2. |
The algorithm efficiently searches the combined space of contact positions and forces for the dexterous robotic hand, seeking the configuration that minimizes the objective function while satisfying all constraints via penalty functions or direct constraint handling methods.
To validate the proposed minimum-normal-force planning algorithm for the dexterous robotic hand, we conducted simulation analyses. First, we compared our method with an existing multi-objective optimization approach from the literature. Under identical conditions (external wrench, friction coefficient \(\mu=0.3\), sphere radius), the literature method yielded a set of contact forces. Our algorithm produced a different force distribution, as shown in the comparison below:
| Method / Finger | \(f_x\) (N) | \(f_y\) (N) | \(f_z\) (N) |
|---|---|---|---|
| Literature (Finger 1) | -0.7221 | 0.8333 | 7.7362 |
| Our Method (Finger 1) | -1.1633 | -0.7756 | 3.6505 |
| Literature (Finger 2) | 0.7115 | 0.8333 | 5.2178 |
| Our Method (Finger 2) | 3.9615 | -0.4803 | 1.4748 |
| Literature (Finger 3) | 0.0106 | 0.8333 | 3.9686 |
| Our Method (Finger 3) | 1.1990 | -1.0857 | 1.5770 |
The results are striking. Our minimum-normal-force planning for the dexterous robotic hand achieved reductions in the critical normal force components (\(f_z\)) of 52.8%, 71.7%, and 60.2% for fingers 1, 2, and 3, respectively, compared to the prior method. While tangential forces (\(f_x, f_y\)) saw some increase in magnitude, the total force magnitude at each contact decreased significantly. This demonstrates the effectiveness of our approach in substantially lowering the “squeeze” force applied by the dexterous robotic hand, which is crucial for handling delicate objects.
In a second, more comprehensive simulation, we planned a grasp for a 0.1 kg spherical object (radius 1 mm) subjected only to gravity. The adaptive DE algorithm was run for 300 generations. The optimal contact forces and corresponding contact points found by the algorithm are listed below:
| Contact Point (Finger) | \(f_x\) (N) | \(f_y\) (N) | \(f_z\) (N) | X (mm) | Y (mm) | Z (mm) |
|---|---|---|---|---|---|---|
| 1 | 0.1077 | 0.6514 | 0.1600 | 0.7172 | 0.1546 | 0.3225 |
| 2 | -0.7256 | -0.4429 | 0.3304 | 0.9174 | -0.5802 | 0.1788 |
| 3 | -0.1226 | 0.2731 | 0.2875 | -0.0762 | 0.5432 | -0.5666 |
The convergence plot of the objective function (sum of squared normal forces) showed a steady decline, confirming the algorithm’s effectiveness. The plots of individual force components over generations revealed that the normal forces started from low initial values and were actively minimized, while the tangential forces adjusted to meet the equilibrium and friction constraints. To evaluate the quality of the obtained grasp from a stability robustness perspective, we employed the common \(\delta\) metric, defined as the minimum singular value of the grasp matrix \(\mathbf{G}\):
$$
\delta = \sigma_{min}(\mathbf{G})
$$
A larger \(\delta\) indicates a greater distance to instability, meaning the grasp can resist larger disturbance wrenches before losing force closure. Analyzing the \(\delta\) metric across the search space near the optimized contact points confirmed that our algorithm converged to regions with high grasp stability indices. Specifically, the second contact point was located in an area with the highest \(\delta\) value (optimal stability), while the first and third points were in regions of high, though slightly lower, stability. This demonstrates that the algorithm for the dexterous robotic hand not only minimizes crushing forces but also inherently favors contact configurations that are robust against uncertainties in control and friction.
This article presented a force-optimized grasp planning strategy for dexterous robotic hands, focusing on minimizing the normal components of contact forces. By formulating the problem as a constrained optimization with force closure, friction cone, and actuator limits as constraints, and employing an adaptive Differential Evolution solver, the method efficiently computes both optimal contact locations and the minimal set of forces required for a stable grasp. The simulation results validated the approach, showing significant reductions in normal forces compared to other methods while maintaining or improving grasp robustness as measured by the minimum singular value metric. The proposed framework is general and practical: it can be adapted to objects of various shapes by modifying the surface constraint, can incorporate different stiffness profiles across an object’s surface via the weighting matrix \(\mathbf{Q}\), and is independent of the specific kinematic structure or number of fingers of the dexterous robotic hand. This makes it a valuable tool for enabling robots to perform secure yet gentle manipulation tasks in real-world applications.